Problem 27 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sin \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The exact value is \(\frac{\sqrt{3}}{2}\), approximately 0.866.

Step 1: Understanding the Problem

We need to evaluate the trigonometric function \(\sin \frac{\pi}{3}\). The goal is to find both its exact value and a decimal approximation if necessary.

Step 1: Recall Known Values

The value of the sine function at \(\frac{\pi}{3}\) is a well-known trigonometric identity. Recall that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). This is an exact value and is derived from common angle values in trigonometry.

Step 2: Determine If the Value Is Irrational

The expression \(\frac{\sqrt{3}}{2}\) is irrational because \(\sqrt{3}\) is an irrational number. This means the exact value includes an irrational component.

Step 3: Find a Decimal Approximation

To find a decimal approximation, calculate \(\frac{\sqrt{3}}{2}\) using a calculator. This gives us approximately \(0.86602540378\).

Key Concepts

Exact Values

In trigonometry, exact values are crucial for understanding and solving problems with precision. An exact value of a trigonometric function refers to its precise mathematical expression without any approximations. These values often arise from well-known angles like

• 30° (\(\frac{\pi}{6}\)
• 45° (\(\frac{\pi}{4}\)
• 60° (\(\frac{\pi}{3}\)

which have commonly agreed-upon trigonometric values. For example, the exact value of \(\sin \frac{\pi}{3}\) is \(\frac{\sqrt{3}}{2}\). This value is derived from the properties of a 30-60-90 triangle where, traditionally, such fractions are used to represent precise values.
Using exact values allows us to maintain mathematical accuracy in calculations, an important attribute in fields like physics and engineering. When working with trigonometric functions, it's critical to recognize these key angles and their trigonometric values.

Irrational Numbers

Understanding irrational numbers plays an important role in grasping trigonometry and its associated functions. An irrational number is one that cannot be expressed as a simple fraction. These numbers continue infinitely in a non-repeating pattern. A classic example of an irrational number found frequently in trigonometry is \(\sqrt{3}\).
When the function \(\sin \frac{\pi}{3}\) was determined to be \(\frac{\sqrt{3}}{2}\), the root of \(3\) is irrational, which makes the entire expression irrational.
Every time you deal with an irrational number, understand that it provides an element of exactness which decimal approximations can't match. While exact values include irrational numbers, their traditional form helps keep the precision intact during mathematical operations. Recognizing these numbers is important to prevent any errors in computations, especially when precision is essential.

Decimal Approximation

While exact values offer precision, working with them on a practical level often requires a decimal form. Decimal approximation helps to simplify and make sense of these numbers in real-world applications.
For example, \(\frac{\sqrt{3}}{2}\) can be roughly represented as \(0.86602540378\) using a calculator. This is immensely helpful in providing a "digestible" number to use in computations, data interpretation, and everyday applications.

• Keep the number of decimal places consistent based on your need for precision.
• Be aware that rounding can affect calculations, especially in iterative processes.

Using decimal approximations is commonplace in fields requiring approximate answers quickly, like construction and design.
However, always start with exact values when solving problems to ensure every successive step in your calculations maintains accuracy.